THEORY AND DESIGN OF WAVE-FILTERS 33 



whence 



{A aa +A bb ) V q -A ab (V g -i+ V q+l ) =0 (2) 



which is the Difference Equation of Propagation. 

 Letting 



V^Me-^ + Nei 1 , (3) 



equation (2) becomes 



{A aa + A bb - 2A ab cosh Y)V q = 



which gives, for all values of V q , 



, ^ A a a~\~A bb 



cosh r = — p— ; 



2A ab 



Since equations (1) when combined give 



I q = \{A a a-A bb )V q + hA ab (V q ^ l -V q+l ), 

 we have upon the substitution of (3) 

 T 1 „ _ _or 1 AT ?r 



Kn K 



wherein the characteristic impedances K a and K b , as defined by the 

 equation, are 



J. 



K a 



1 



= A a vsmhr±%(Aaa — Abb). 



In terms of the admittances then 



i t, A. aa ~T A bb 



cosh T = — — ; » 



2A ab 



and (4) 



K a 

 K b 



'. a \ =1 / A aa +Abb \ S j / 2A ab \ 2 / A aa -A bb \ 



: b \ 2 \A aa A bb -AlJ ) \ U aa +^ W / ^Uoa + ^bJ 



These formulae can readily be expressed in terms of the impedances 

 Z aa , Z bb , and Z ab ; or in terms of the three star-connected (7") or three 

 delta-connected (II) impedances which may represent the section. 



Another general formula for the propagation constant which is 

 sometimes convenient may be derived as follows. Assume that the 

 recurrent structure is open-circuited at the junction q; then in (1) 

 I g = 0, so that 



Vq- l _ Abb_ _. j^ 

 V q A ab Vab' 



and V g+l A_aa _ 1_ 



Vq A ab Vba 



